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We consider the following perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 +V(x)$ on the real line. The potential $V(x)$ is a real - valued function ofshort range type. We study the equivalence of classical homogeneous Sobolevtype spaces $\dot{H}^s_p$, $p \in (1,\infty)$ and the corresponding perturbedhomogeneous Sobolev spaces associated with the perturbed Hamiltonian. It isshown that the assumption zero is not a resonance guarantees that the perturbedand unperturbed homogeneous Sobolev norms of order $s = \gamma - 1 \in [0,1/p)$are equivalent. As a corollary, the corresponding wave operators leaveclassical homogeneous Sobolev spaces of order $s \in [0,1/p)$ invariant.

Hardy inequality and fractional Leibnitz rule for perturbed Hamiltonians on the line